Deflation and Residuation for Class Formation

In the present paper, recalling briefly a deflation map and a residuation map in the cohomology theory of finite groups, we shall see that a certain pair of isomorphisms of Tateʹs main theorem for class formations establishes a correspondence of the associated residuation map to the associated deflation map (Theorem 1). This result, supplementary to the results of Artin and Tate, will further provide us with a commutative diagram for Galois cohomology groups, which just generalizes a fundamental commutative diagram for number knots (Theorem 2).